Tables and Formulas - Digital Signal Processing: An Introduction with MATLAB and Applications - page 315

Butterworth DC gain: G dc = G m
Magnitude response of Chebychev-I LPF:

,
1 þ e 2 C n

s

x
x c

j H ð x Þj¼ G m

where C o (x) = 1, C 1 (x) = x, C n+1 (x) = 2xC n (x) - C n-1 (x).
Chebychev DC gain:

G m n odd

G m =

p

G dc ¼

1 þ e 2

n even

Matched filter: h(t) = s(T - t).
Inverting and non-inverting amplifiers:

Z 2 Z 2

Z 1 Z 1

v − v −

_ _

X

Y Y

v + v +

+ + + X + + +

Y / X = − Z 2 / Z 1
Inverting Amplifier

Y / X = 1 + Z 2 / Z 1
Non−inverting Amplifier

Hilbert Transformer (HT): H(f) =-jsgn(f) with h(t) = 1/(p t).
Hilbert Transform of x(t) is: x ð t Þ¼ x ð t Þ h ð t Þ or: X ð f Þ¼ j sgn ð f Þ X ð f Þ .
The analytic associate of x(t) is: z ð t Þ¼ x ð t Þþ j x ð t Þ
Digital HT:

H ð e jX Þ¼ j sgn ð X Þ;j X j \p; h ð n Þ¼ p sin 2 ð n p = 2 Þ

n 6¼ 0
elsewhere

;

n

0

Bilinear Transform: s ) z 1
z þ 1

with X ¼ 2 tan 1 ð x Þ¼ 2 tan 1 ð 2pf Þ:

Impulse invariant transform:

H a ð s Þ¼ X ) H ð z Þ¼ T s X

M M

c m
s p m

z
z z m

z m ¼ e p m T s : c m where

m ¼ 1 m ¼ 1

Digital domain frequency: X ¼ 2p v ¼ 2p f = f s ¼ x = f s ¼ xT s
Sinusoidal PLL system equation: / ð k Þ¼ h ð k Þ x 0 P k 1 y ð i Þ

i ¼ 0

Sinusoidal PLL: 1st-order system equation:

/ ð k þ 1 Þ¼ / ð k Þ K 2 sin ½ / ð k Þþ K o
where K o ¼ 2p ð x x o Þ= x o and K 2 xG 1 A :

If K 1 xG 1 A and W ¼ x o = x ; then K 2 ¼ K 1 ð x o = x Þ¼ K 1 = W :